Theory and Applications of Fractional Order SystemsView this Special Issue
Research Article | Open Access
Finite Series Representation of the Inverse Mittag-Leffler Function
The inverse Mittag-Leffler function is valuable in determining the value of the argument of a Mittag-Leffler function given the value of the function and it is not an easy problem. A finite series representation of the inverse Mittag-Leffler function has been found for a range of the parameters and ; specifically, for and for . This finite series representation of the inverse Mittag-Leffler function greatly expedites its evaluation and has been illustrated with a number of examples. This represents a significant advancement in the understanding of Mittag-Leffler functions.
The Mittag-Leffler function is defined by the power series  While the argument and the parameters and can in general be complex provided Re , in this work , , and will be restricted to those values most commonly found in physical problems; namely, the argument will be restricted to real numbers and and will be restricted to positive real numbers. The Mittag-Leffler function is a generalization of the exponential function and arises frequently in the solutions of differential and/or integral equations of fractional (noninteger) order in much the same way as the exponential function appears in solutions of differential equations of integer order. Thus, Mittag-Leffler functions play a fundamental role in the theory of fractional differential equations. Consequently, books devoted to the subject of fractional differential equations (i.e., Podlubny , Kilbas et al. , and Diethelm ) all contain sections on the Mittag-Leffler functions. In addition to their inherent mathematical interest, Mittag-Leffler functions are also important in theoretical and applied physics and all the sciences (i.e., Hilfer , Mainardi , and Magin ). The works of Mainardi and Gorenflo , Magin , Berberan-Santos , Gupta and Debnath , and Haubold et al.  are a few of the numerous articles also worth noting.
The inverse Mittag-Leffler function is defined as the solution of (2)  Despite the inherent importance of Mittag-Leffler functions in fractional differential equations, with the wealth of analytical information about , the inverse has been largely unexplored. The one exception is the excellent work of Hilfer and Seybold  who have determined its principal branch numerically.
The power series representation of any Mittag-Leffler function can be inverted yielding an infinite series for the inverse. However, these infinite series are slow to converge and terminating the series always introduces error which is hard to evaluate. This present work identifies regions in the domain of and where the inverse of the Mittag-Leffler function can be written as a finite series. This represents the first time the inverse Mittag-Leffler function has been written as a finite series as opposed to an infinite series which greatly expedites its evaluation. Before deriving these expressions for the inverse Mittag-Leffler function, a brief review of the theory of power series and their inverses is in order.
Consider the convergent series which expresses the function in terms of powers of with the corresponding coefficients given by The inversion of the function requires only the sole assumption that . That is, there exists one and only one function which represents the inverse of the , which is expressible by a convergent power series of the form  The process of finding the series expansion for is called reversion of the series. The coefficients can be determined in terms of the coefficients by substituting (3) into (4) and equating coefficients of like powers of on both sides of the equation yielding The coefficients can be found in the literature [15–17]. An explicit expression for the coefficients can be derived using the Lagrange inversion theorem. If is analytic at and , then the inverse of exists and is analytic about . Furthermore, if , the Lagrange inversion theorem gives the Taylor series expansion of the inverse function as  The coefficients are determined by comparing (6) and (4) yielding Substituting from (3) yields Factoring out in (8) and defining yields Using the multinomial expansion and performing the required differentiation yields the desired result  where and the numbers are nonnegative integers and the summation extends over all partitions of . For example, contains 5 terms since the Diophantine equation has 5 integer solutions or partitions. The number of partitions for is 42; for there are 204226 partitions and for the number of partitions is 190569292. Consequently, the explicit tabulation of the full expression for the coefficients rapidly becomes a rather tedious task. Nevertheless, the coefficients are given in Table 1. An equivalent expression for the general term in the reversion of series is given in a different form by McMahon .